Logarithms: Introduction

Subject: Business Math

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Overview

The logarithm of a number is the exponent factor, to which the base, another fixed number, must be raised to produce that number. The logarithms of the very same number vary from each other, in case of different bases. Common logarithms are the logarithms with base 10. The logarithm of a number consists of 2 parts; the 1st part is called Characteristic, whereas the 2nd part is called Mantissa.
Logarithms: Introduction

LOGARITHMS

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Introduction:

The logarithm of a number is the index of a power, to which an already given positive base must be raised to equal the number. Introduced in the early 17th century by John Napier, the logarithm of a number is the exponent factor, to which the base, another fixed number, must be raised to produce that number.
For example;

The base 100 logarithm of 10000 is 2, as 100 to the power 2 is 10000,
[10000 = 100 x 100 = 100²]
i.e. 10000 = 100² or 2 = log10010000

Likewise,

The base 2 logarithm of 64 is 6, as 2 to the power 6 is 64,
[64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
i.e. 64 = 26or 6 = log6464

Also,

64 = 4³ or 3 = log464

64 = 64¹ or 1 = log6464

Therefore, the logarithms of the very same number vary from each other, in case of different bases.

Laws of Logarithms:

The following rules are applicable for any positive numbers x, y and e:

  1. logee= 1 which follows the fact that e¹ = e.
  2. loge1= 0 which follows the fact that e0 = 1.

For the following rules, assume x>0 and y>0.

  1. logexy= y logex
  2. loge(xy) = logex + logey
  3. loge (x/y) = logex - logey

Examples:

Find the value of:

  1. log8512
  2. log√216

Solution:

  1. Let, x = log8512
    or, 8x = 512
    or, 8x= 8³
    so, x = 3

    Alternatively,

    log8 512 = log8
    = 3 log88.................. (logexy= y logex)
    = 8 x 1..................... (logee = 1)
    = 8

  2. Let, x = log√216

    or, √2x = 24

    or, √2x= [(√2)²]

    or, √2x= (√2)²

    so, x = 8

Common Logarithms:

Common logarithms are the logarithms with base 10. Almost all the numerical computations use the base 10, which is why, in the absence of the base, it is understood that the base is 10.
Such as;

10¹ = 10 or log1010 = 1
10² = 100 or log1010 = 2
10³ = 1000 or log1010 = 3
104 = 10000 or log1010 = 4 and so on.

From the above examples, it can be observed that the logarithm of a number between 10 & 100 will be between 1 & 2. Therefore, it is equal to 1+a, where a is a positive proper fraction. Similarly, logarithms of numbers 100 & 1000 and 1000 & 10000 will be between 2 & 3 (equal to 2+b where b is a positive proper fraction) and 3 & 4 (equal to 3+c where c is a positive proper fraction) respectively and so on.

Again,

10-¹ = 0.1 or log100.1 = -10.1 = -1

10-² = 0.01 or log100.01 = -2 an so on.

Above examples show that the logarithm of a number between 0.1 & 0.01 will lie between -1 & -2. Hence, it is equal to -2+a where a is a positive proper fraction, which provides us with following conclusions:

-The logarithm of a number consists of 2 parts-

  1. An integer (positive, negative or zero)
  2. A positive proper fraction

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Here, the 1st part is called Characteristic, whereas the 2nd part is called Mantissa. Therefore, a logarithm’s characteristic is an integer while its mantissa is a positive proper fraction.

Rules for finding Characteristic:

To find characteristic, there are 2 ways. They are:

  1. For numbers greater than or equal to one:

    The characteristic of numbers greater than or equal to one is one less than the number of digits contained in its integral part.” This is the rule that applies in the case of numbers that are greater than or equal to one.

    Since,

    10° = 1 or, log101 = 0

    10¹ = 10 or, log1010 = 1

    The above observation shows that that the value of the logarithm of numbers between 1 & 10 is 0+a, where a is the positive proper fraction. Therefore, 0 is the characteristic of numbers lying between 1 and 10.

    Likewise,

    10² = 100 or, log10100 = 2

    Hence, for the numbers between 10 & 100, the logarithm value is 1+a, where a is the positive proper fraction. Therefore, the characteristic of numbers lying between 1 and 10 is 1.

    Similarly, characteristic of numbers lying between 100 & 1000 is 2 and so on goes the process.

  1. For numbers less than one:
    The characteristic of numbers less than unity is negative and is one more than the number of ciphers immediately after the decimal point before any significant (i.e. non-zero) figure.

    Since,

    10° = 1 or, log101 = 0

    10-¹ = 0.1 or, log100.1 = -1

    Therefore, the value of the logarithm of numbers between 0.1 & 1 is -1+a, where a is the positive proper fraction. Thus, it gives the characteristic -1 between the numbers 0.1 & 1.

    Likewise,

    10-² = 0.01 or, log100.01 = -2

    So, the value of the logarithm of numbers between 0.1 & 0.01 is -2+a, where a is the positive proper fraction, thus giving the characteristic -2 and so on.

    Here, it is notable that in the case of a number without a cipher, immediately there is -1 after decimal characteristic of its logarithm. In case a number has one cipher, immediately there is -2 after decimal characteristic of its logarithm and so goes on.

(Tamang, Pant, & G.C, 2016)



Bibliography

Tamang, G., Pant, N., & G.C, P. B. (2016). Business Mathematics. Putalisadak: Asmita Publication.

Things to remember

Logarithms

  • Introduction
  • Laws of Logarithms
  • Common Logarithms
  • Rules for finding Characteristic
  • It includes every relationship which established among the people.
  • There can be more than one community in a society. Community smaller than society.
  • It is a network of social relationships which cannot see or touched.
  • common interests and common objectives are not necessary for society.

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